Assumption REF  shows that \(\mathcal {C}\)  is a compact convex set with diameter \(R\) . Assumption REF  indicates that the constraint functions \(G_i(\cdot ,\xi _k)\)  are bounded over \(\mathcal {C}\) . This assumption is a bit restrictive, but it is required in the analysis of almost all existing stochastic methods for solving problem (REF ) [1]}, [2]}, [3]}, [4]}. Assumption REF  requires that the stochastic subgradients \(v_i(\cdot ,\xi _k)\)  are bounded over \(\mathcal {C}\) . Assumption REF  is a standard Slater's condition for optimization problem with functional constraints.
